设$\mathbb{D}$是复平面$\mathbb{C}$上的单位圆盘, $H(\mathbb{D})$是$\mathbb{D}$上的解析函数空间. Bloch空间$\mathcal{B}=\mathcal{B}(\mathbb{D})$定义为

$\mathcal{B}=\{f\in H(\mathbb{D}):\ \|f\|_{b}=\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})|f'(z)|<\infty\}.$

Zygmund空间$\mathcal{Z}=\mathcal{Z}(\mathbb{D})$定义为

$\mathcal{Z}=\{f\in H(\mathbb{D}):\ \|f\|_{z}=\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})|f''(z)|<\infty\}.$

$\alpha$-Bloch空间$\mathcal{B}^{\alpha}(\mathbb{D})$定义为

$\mathcal{B}^{\alpha}=\{f\in H(\mathbb{D}):\ \|f\|_{b^{\alpha}}=\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})^{\alpha}|f'(z)|<\infty, \ \alpha>0\}.$

$\alpha$-Zygmund空间$\mathcal{Z}^{\alpha}=\mathcal{Z}^{\alpha}(\mathbb{D})$定义为

$\mathcal{Z}^{\alpha}=\{f\in H(\mathbb{D}):\ \|f\|_{z^{\alpha}}=\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})^{\alpha}|f''(z)|<\infty\ \alpha>0\}.$

当$\alpha=1$, 则$\mathcal{B}^{\alpha}$和$\mathcal{Z}^{\alpha}$分别为$\mathcal{B}$和$\mathcal{Z}$.

作为Bloch空间的推广, 近来, Ramos Fernándz在文献[1]中利用Young函数定义了Bloch-Orlicz空间.设$\varphi:[0, \infty)\rightarrow[0, \infty)$为一个严格递增的凸函数, 且$\varphi(0)=0$, $\lim\limits_{t\rightarrow\infty}\varphi(t)=\infty$.如果对于某个只依赖于$f\in H(\mathbb{D})$的$\lambda>0$,

$\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})\varphi(\lambda|f'(z)|)<\infty, $

称$f$属于Bloch-Orlicz空间, 记为$\mathcal{B}^{\varphi}=\mathcal{B}^{\varphi}(\mathbb{D})$.显然, $\mathcal{B}^{\varphi}$为$F$-空间, 且若$\varphi(t)=t, t\geq0$, $\mathcal{B}^{\varphi}$即为Bloch空间$\mathcal{B}$, 由于$\varphi$是凸函数, 不难验证Minkowski泛函

$\|f\|_{b^{\varphi}}=\inf\bigg\{k>0:S_{\varphi}\bigg(\frac{f'}{k}\bigg)\leq1\bigg\}$

定义了$\mathcal{B}^{\varphi}$上的一个半范数, 即Luxemburg半范数, 其中

$S_{\varphi}(f)=\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})\varphi(|f(z)|).$

事实上, 可以证明$\mathcal{B}^{\varphi}$在范数$\|f\|_{\mathcal{B}^{\varphi}}=|f(0)|+\|f\|_{b^{\varphi}}$下是一个Banach空间且

$S_{\varphi}\bigg(\frac{f'}{\|f\|_{\mathcal{B}^{\varphi}}}\bigg)\leq1.$

利用上式可以证明Bloch-Orlicz空间$\mathcal{B}^{\varphi}$等距同构于$\mu$-Bloch空间, 其中

$\mu(z)=\frac{1}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}, z\in\mathbb{D}.$

因此, 对于任何$f\in\mathcal{B}^{\varphi}\backslash\{0\}$, 有

$\|f\|_{\mathcal{B}^{\varphi}}=|f(0)|+\sup\limits_{z\in\mathbb{D}}\mu(z)|f'(z)|.$

在文献[2]中, 作者定义了Zygmund-Orlicz空间.如果对于某个只依赖于$f\in H(\mathbb{D})$的$\lambda>0$,

$\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})\varphi(\lambda|f''(z)|)<\infty, $

则称$f$属于Zygmund-Orlicz空间, 记为$\mathcal{Z}^{\varphi}=\mathcal{Z}^{\varphi}(\mathbb{D})$.类似于Bloch-Orlicz空间, 利用$\varphi$的凸性, 可以验证Minkowski泛函

$\|f\|_{z^{\varphi}}=\inf\bigg\{k>0:S_{\varphi}\bigg(\frac{f''}{k}\bigg)\leq1\bigg\}$

为$\mathcal{Z}^{\varphi}$定义了一个半范数.此外还可证明$\mathcal{Z}^{\varphi}$在范数$\|f\|_{\mathcal{Z}^{\varphi}}=|f(0)|+|f'(0)|+\|f\|_{z^{\varphi}}$下是一个Banach空间且

$S_{\varphi}\bigg(\frac{f''}{\|f\|_{\mathcal{Z}^{\varphi}}}\bigg)\leq1.$

利用上式可以证明Zygmund-Orlicz空间$\mathcal{Z}^{\varphi}$等距同构于$\mu$-Zygmund空间, 其中

$\mu(z)=\frac{1}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}, z\in\mathbb{D}.$

因此, 对于任何$f\in\mathcal{Z}^{\varphi}\backslash\{0\}$, 有

$\|f\|_{\mathcal{Z}^{\varphi}}=|f(0)|+|f'(0)|+\sup\limits_{z\in\mathbb{D}}\mu(z)|f''(z)|.$

设$\phi$为$\mathbb{D}$的解析自映射, $g:\mathbb{D}\rightarrow\mathbb{C}$为一个解析映射,

$\big(C_{\phi}^{g}f\big)(z)=\int_{0}^{z}f'(\phi(\xi))g(\xi)d\xi, z\in\mathbb{D}, $

称$C_{\phi}^{g}$为广义复合算子.当$g=\phi'$时,

$\big(C_{\phi}^{g}f\big)(z)=\int_{0}^{z}f'(\phi(\xi))\phi'(\xi)d\xi=f(\phi(z))-f(\phi(0)), $

由于$f(\phi(0))$为一个常数, 故$C_{\phi}^{g}$实质上就是复合算子$C_{\phi}$.因此, $C_{\phi}^{g}$可以看作复合算子的推广, 见文献[3-9].在文献[2]中, 作者研究了从Zygmund空间到Bloch-Orlicz空间和Zygmund-Orlicz空间的广义复合算子的有界性和紧性, 本文研究从$\alpha$-Zygmund空间到$\mathcal{B}^{\varphi}$和$\mathcal{Z}^{\varphi}$的广义复合算子的有界性和紧性.由于当$\alpha=1$时, $\alpha$-Zygmund空间为Zygmund空间$\mathcal{Z}$, 故考虑当$\alpha\neq1$时, $C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$和$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$的有界性和紧性.文中的$C$是一个正常数, 在不同地方可以表示不同的值.

在引理2.1中, 当$\alpha=1$时, 可参见文献[10], 其它的情况也可类似证明, 见文献[11].利用文献[12]中命题3.11的方法, 不难证明下列引理2.2.

引理2.1  对于任何$f\in\mathcal{Z}^{\alpha}$, $\alpha>0$, 我们有

(ⅰ)当$0<\alpha<1$时, $|f'(z)|\leq\frac{2}{1-\alpha}\|f\|_{\mathcal{Z}^{\alpha}}$;

(ⅱ)当$\alpha=1$时, $|f'(z)|\leq\log\big(\frac{e}{1-|z|^{2}}\big)\|f\|_{\mathcal{Z}}$;

(ⅲ)当$\alpha>1$时, $|f'(z)|\leq\frac{2}{\alpha-1}\frac{\|f\|_{\mathcal{Z}^{\alpha}}}{(1-|z|^{2})^{\alpha-1}}$.

引理2.2  设$g\in H(\mathbb{D})$, $\phi$为$\mathbb{D}$的解析自映射.令$X=\mathcal{Z}^{\alpha}$, $Y=\mathcal{B}^{\varphi}$或$\mathcal{Z}^{\varphi}$, 则$C_{\phi}^{g}:X\rightarrow Y$是紧算子当且仅当$C_{\phi}^{g}:X\rightarrow Y$是有界算子, 且对于$\mathcal{Z}^{\alpha}$中在$\mathbb{D}$的紧子集上一致收敛于零的任意有界序列$\{f_{n}\}_{n\in\mathbb{N}}$, 都有$\|C_{\phi}^{g}f_{n}\|_{Y}\rightarrow0, n\rightarrow\infty$.

引理2.3  设$0<\alpha<1$, $\{f_{n}\}_{n\in\mathbb{N}}$为$\mathcal{Z}^{\alpha}$中的任意有界序列且在$\mathbb{D}$的紧子集上一致收敛于零, $n\rightarrow\infty$.则

$\lim\limits_{n\rightarrow\infty}\sup\limits_{z\in\mathbb{D}}|f'_{n}(z)|=0.$

证 设$M=\sup\limits_{n}\|f_{n}\|_{\mathcal{Z}^{\alpha}}<\infty$.任取$\epsilon>0$, 存在$0<\eta<1$使得$(1-\eta)^{1-\alpha}<\epsilon$.若$\eta<|z|<1$, 则

$\begin{eqnarray*} \bigg|f'_{n}(z)-f'_{n}\bigg(\frac{\eta}{|z|}z\bigg)\bigg|&=&\bigg|\int^{1}_{\frac{\eta}{|z|}}z f''_{n}(z\eta)d\eta\bigg|= \bigg|\int^{1}_{\frac{\eta}{|z|}}(1-|z\eta|^{2})^{\alpha}\frac{f''_{n}(z\eta)z}{(1-|z\eta|^{2})^{\alpha}}d\eta\bigg|\\ &\leq&M\int^{1}_{\frac{\eta}{|z|}}\frac{|z|}{(1-|z\eta|^{2})^{\alpha}}d\eta=M\int^{|z|}_{\eta}\frac{d\xi}{(1+\xi)^{\alpha}(1-\xi)^{\alpha}}\\ &\leq&M\int^{|z|}_{\eta}(1-\xi)^{-\alpha}d\xi\leq\frac{M}{1-\alpha}(1-\eta)^{1-\alpha}\leq\frac{M}{1-\alpha}\epsilon. \end{eqnarray*} $

故

$\begin{eqnarray*}\sup\limits_{\eta<|z|<1}|f'_{n}(z)|\leq\frac{M}{1-\alpha}\epsilon+\sup\limits_{|z|=\eta}|f'_{n}(z)|. \end{eqnarray*}$

由于$\{f_{n}\}$在$\mathbb{D}$的紧子集上一致收敛于零, 则由Cauchy估计, $\{f'_{n}\}$也在$\mathbb{D}$的紧子集上一致收敛于零, 有

$\begin{eqnarray*} \lim\limits_{n\rightarrow\infty}\sup\limits_{z\in\mathbb{D}}|f'_{n}(z)|\leq \lim\limits_{n\rightarrow\infty}\sup\limits_{z\in\mathbb{D}}\bigg(\frac{M}{1-\alpha}\epsilon+2\sup\limits_{|z|\leq\eta}|f'_{n}(z)|\bigg)=\frac{M}{1-\alpha}\epsilon. \end{eqnarray*} $

因此$\lim\limits_{n\rightarrow\infty}\sup\limits_{z\in\mathbb{D}}|f'_{n}(z)|=0$.

本部分我们给出广义复合算子$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$的有界性和紧性的特征.

定理3.1  设$g\in H(\mathbb{D})$, $\phi$为$\mathbb{D}$的解析自映射, $0<\alpha<1$.则下列命题等价:

(ⅰ) $C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是紧算子;

(ⅱ) $C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是有界算子;

(ⅲ)

$\begin{eqnarray} k_{1}=\sup\limits_{z\in\mathbb{D}}\frac{|g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}<\infty. \end{eqnarray}$

(3.1)

证 (ⅰ)$\Rightarrow$(ⅱ)由于紧算子是有界算子, 显然成立.

(ⅱ)$\Rightarrow$(ⅲ)设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是有界算子, 则对于所有$f\in\mathcal{Z}^{\alpha}$, 都存在一个常数$C$使得$\|C_{\phi}^{g}f\|_{\mathcal{B}^{\varphi}}\leq C\|f\|_{Z^{\alpha}}$.取函数$f(z)=z\in \mathcal{Z}^{\alpha}$, 显然有$\|f\|_{\mathcal{Z}^{\alpha}}=1$, 则

$\begin{eqnarray*} S_{\varphi}\bigg(\frac{(C_{\phi}^{g}f)'(z)}{C\|f\|_{\mathcal{Z}^{\alpha}}}\bigg) =S_{\varphi}\bigg(\frac{g(z)}{C}\bigg) =\sup\limits_{z\in \mathbb{D}}(1-|z|^{2})\varphi\bigg(\frac{|g(z)|}{C}\bigg)\leq1. \end{eqnarray*}$

(ⅲ)$\Rightarrow$(ⅰ)设$k_{1}<\infty$, 令$\{f_{n}\}_{n\in\mathbb{N}}$为$\mathcal{Z}^{\alpha}$中的任意有界序列, 且在$\mathbb{D}$的紧子集上一致收敛于零, $n\rightarrow\infty$.由于$(C_{\phi}^{g}f_{n})(0)=0$, 故

$\begin{eqnarray*} \|C_{\phi}^{g}f_{n}\|_{\mathcal{B}^{\varphi}} =\sup\limits_{z\in\mathbb{D}}\frac{|(C_{\phi}^{g}f_{n})'(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})} =\sup\limits_{z\in\mathbb{D}}\frac{|f'_{n}(\phi(z))||g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})} \leq k_{1}\sup\limits_{z\in\mathbb{D}}|f'_{n}(\phi(z))|. \end{eqnarray*}$

由引理2.3可知$\lim\limits_{n\rightarrow\infty}\sup\limits_{z\in\mathbb{D}}|f'_{n}(\phi(z))|=0$.因此, $\lim\limits_{n\rightarrow\infty}\|C_{\phi}^{g}f_{n}\|_{\mathcal{B}^{\varphi}}=0$, 由引理2.2可知$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是紧算子.

定理3.2  设$g\in H(\mathbb{D})$, $\phi$为$\mathbb{D}$的解析自映射, $\alpha>1$.则$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是有界算子当且仅当

$\begin{eqnarray} k_{2}=\sup\limits_{z\in\mathbb{D}}\frac{|g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha-1}}<\infty. \end{eqnarray}$

(3.2)

证 假设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是有界的.利用函数$f(z)=z\in \mathcal{Z}^{\alpha}$, 类似于定理3.1的证明, 则$k_{1}<\infty$.对于任何$b, z\in\mathbb{D}$, 令$h_{b}(z)=\frac{(1-|b|^{2})^{2}}{(1-\overline{b}z)^{\alpha}}$, 则

$\begin{eqnarray*} \sup\limits_{z\in\mathbb{D}}(1-|z|^{2})^{\alpha}|h''_{b}(z)| &\leq&\sup\limits_{z\in\mathbb{D}}(1+|z|)^{\alpha}(1-|z|)^{\alpha}\frac{\alpha(\alpha+1)(1+|b|)^{2}(1-|b|)^{2}|b|^{2}}{(1-|z|)^{\alpha}(1-|b|)^{2}}\\ &\leq&4\cdot2^{\alpha}\cdot\alpha(\alpha+1)<\infty, \end{eqnarray*}$

则$h_{b}\in\mathcal{Z}^{\alpha}$.令$b=\phi(\omega), \omega\in\mathbb{D}$且使得$\frac{1}{2}<|\phi(\omega)|<1$, 直接计算可得

$\begin{eqnarray} h'_{\phi(\omega)}(\phi(\omega))=\frac{\alpha\overline{\phi(\omega)}}{(1-|\phi(\omega)|^{2})^{\alpha-1}}. \end{eqnarray} $

(3.3)

由$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$的有界性知, 存在一个常数$C$, 使得$\|C_{\phi}^{g}h_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C$, 则由(3.3) 式,

$\begin{eqnarray*} 1&\geq& S_{\varphi}\bigg(\frac{(C_{\phi}^{g}h_{\phi(\omega)})'(z)}{C}\bigg)\geq \sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}(1-|\omega|^{2})\varphi\bigg(\frac{|(C_{\phi}^{g}h_{\phi(\omega)})'(\omega)|}{C}\bigg)\\ &=&\sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}(1-|\omega|^{2})\varphi \bigg(\frac{\alpha|\phi(\omega)||g(\omega)|}{C(1-|\phi(\omega)|^{2})^{\alpha-1}}\bigg). \end{eqnarray*}$

由此可得

$\begin{eqnarray} \sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}\frac{|g(\omega)|}{\varphi^{-1}(\frac{1}{1-|\omega|^{2}})(1-|\phi(\omega)|^{2})^{\alpha-1}} \leq\sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}\frac{2|\phi(\omega)||g(\omega)|}{\varphi^{-1}(\frac{1}{1-|\omega|^{2}})(1-|\phi(\omega)|^{2})^{\alpha-1}}<\infty. \end{eqnarray}$

(3.4)

利用$k_{1}<\infty$, 有

$\begin{eqnarray} \sup\limits_{|\phi(\omega)|\leq\frac{1}{2}}\frac{|g(\omega)|}{\varphi^{-1}(\frac{1}{1-|\omega|^{2}})(1-|\phi(\omega)|^{2})^{\alpha-1}} \leq k_{1}\bigg(\frac{4}{3}\bigg)^{\alpha-1}<\infty. \end{eqnarray}$

(3.5)

由(3.4) 和(3.5) 式即可得到(3.2) 式.

反之, 假设$k_{2}<\infty$, 对于任何$f\in\mathcal{Z}^{\alpha}\backslash\{0\}$, 由引理2.1 (iii), 有

$\begin{eqnarray*} S_{\varphi}\bigg(\frac{(C_{\phi}^{g}f)'(z)}{C\|f\|_{\mathcal{Z}^{\alpha}}}\bigg) &\leq&\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})\varphi\bigg(\frac{k_{2}\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha-1}|f'(\phi(z))|}{C\|f\|_{\mathcal{Z}^{\alpha}}}\bigg)\\ &\leq&\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})\varphi\bigg(\frac{\frac{2}{\alpha-1}k_{2}}{C}\varphi^{-1}\bigg(\frac{1}{1-|z|^{2}}\bigg)\bigg)\leq1, \end{eqnarray*}$

其中$C\geq\frac{2}{\alpha-1}k_{2}$.故存在一个常数$C$, 使得$\|C_{\phi}^{g}f\|_{\mathcal{B}^{\varphi}}\leq C\|f\|_{\mathcal{Z}^{\alpha}}$, 即$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是有界的.

定理3.3  设$g\in H(\mathbb{D})$, $\phi$为$\mathbb{D}$的解析自映射, $\alpha>1$.则$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是紧算子当且仅当$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是有界算子, 且

$\begin{eqnarray} \lim\limits_{|\phi(z)|\rightarrow1}\frac{|g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha-1}}=0. \end{eqnarray}$

(3.6)

证 假设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是紧算子, 则$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是有界算子.令$\{z_{n}\}_{n\in\mathbb{N}}$是$\mathbb{D}$中的序列且$|\phi(z_{n})|\rightarrow1$, $n\rightarrow\infty$.取函数$ h_{n}(z)=\frac{(1-|\phi(z_{n})|^{2})^{2}}{(1-\overline{\phi(z_{n})}z)^{\alpha}}, $由定理3.2的证明知$\sup\limits_{n\in\mathbb{N}}\|h_{n}\|_{\mathcal{Z}^{\alpha}}<\infty$.同时易见$\{h_{n}\}$在$\mathbb{D}$的紧子集上一致收敛于零, 则由引理2.2, $\|C_{\phi}^{g}h_{n}\|_{\mathcal{B}^{\varphi}}\rightarrow0, n\rightarrow\infty$.则

$\begin{eqnarray*} 1\geq S_{\varphi}\bigg(\frac{(C_{\phi}^{g}h_{n})'(z_{n})}{\|C_{\phi}^{g}f_{n}\|_{\mathcal{B}^{\varphi}}}\bigg) \geq(1-|z_{n}|^{2})\varphi \bigg(\frac{\alpha|\phi(z_{n})||g(z_{n})|}{(1-|\phi(z_{n})|^{2})^{\alpha-1}\|C_{\phi}^{g}h_{n}\|_{\mathcal{B}^{\varphi}}}\bigg), \end{eqnarray*}$

由此可得

$\begin{eqnarray*} \frac{|\phi(z_{n})||g(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha-1}}\leq\frac{1}{\alpha}\|C_{\phi}^{g}h_{n}\|_{\mathcal{B}^{\varphi}}. \end{eqnarray*}$

故

$\begin{eqnarray*} \lim\limits_{|\phi(z_{n})|\rightarrow1}\frac{|g(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha-1}}= \lim\limits_{n\rightarrow\infty}\frac{|\phi(z_{n})||g(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha-1}}=0. \end{eqnarray*}$

反之, 假设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是有界算子且(3.6) 式成立, 则仿照定理3.1, 可得$k_{1}<\infty$, 且对于任意的$\epsilon>0$, 存在$\delta\in(0, 1)$, 使得只要$\delta<|\phi(z)|<1$, 就有

$\begin{eqnarray} \frac{|g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha-1}}<\frac{\alpha-1}{2}\epsilon. \end{eqnarray} $

(3.7)

设$\{f_{n}\}_{n\in\mathbb{N}}$为$\mathcal{Z}^{\alpha}$中的任意有界序列且在$\mathbb{D}$的紧子集上一致收敛于零, 令$K=\{z\in\mathbb{D}:|\phi(z)|\leq\delta\}$, 又$(C_{\phi}^{g}f_{n})(0)=0$, 则由$k_{1}<\infty$及(3.7) 式可得

$\begin{eqnarray*} \|C_{\phi}^{g}f_{n}\|_{\mathcal{B}^{\varphi}} &\leq&\sup\limits_{z\in K}\frac{|g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}|f'_{n}(\phi(z))| +\sup\limits_{z\in \mathbb{D}\backslash K}\frac{\frac{2}{\alpha-1}|g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha-1}}\\ &\leq&k_{1}\sup\limits_{|\omega|\leq\delta}|f'_{n}(\omega)|+\epsilon. \end{eqnarray*}$

由于$\{f_{n}\}$在$\mathbb{D}$的紧子集上一致收敛于零, 则由Cauchy估计, $\{f'_{n}\}$也在$\mathbb{D}$的紧子集上一致收敛于零, 特别地, $\{\omega:|\omega|\leq\delta\}$为$\mathbb{D}$的紧子集, 则有$\lim\limits_{n\rightarrow\infty}\sup\limits_{|\omega|\leq\delta}|f'_{n}(\omega)|=0$, 因此$\lim\limits_{n\rightarrow\infty}\|C_{\phi}^{g}f_{n}\|_{\mathcal{B}^{\varphi}}=0. $由引理2.2可得$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{B}^{\varphi}$是紧算子.

本部分我们给出广义复合算子$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$的有界性和紧性的特征.

定理4.1  设$g\in H(\mathbb{D})$, $\phi$为$\mathbb{D}$的解析自映射, $0<\alpha<1$.则$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界算子当且仅当

$k_{3}=\sup\limits_{z\in\mathbb{D}}\frac{|g'(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}<\infty,$

(4.1)

$k_{4}=\sup\limits_{z\in\mathbb{D}}\frac{|\phi'(z)||g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha}}<\infty.$

(4.2)

证 假设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界算子, 即对于所有$f\in\mathcal{Z}^{\alpha}$, 都存在一个常数$C$使得$\|C_{\phi}^{g}f\|_{\mathcal{Z}^{\varphi}}\leq C\|f\|_{Z^{\alpha}}$.取函数$f(z)=z\in \mathcal{Z}^{\alpha}$, 则

$\begin{eqnarray*} S_{\varphi}\bigg(\frac{(C_{\phi}^{g}f)''(z)}{C\|f\|_{\mathcal{Z}^{\alpha}}}\bigg) =S_{\varphi}\bigg(\frac{g'(z)}{C}\bigg) =\sup\limits_{z\in \mathbb{D}}(1-|z|^{2})\varphi\bigg(\frac{|g'(z)|}{C}\bigg)\leq1, \end{eqnarray*}$

则(4.1) 式成立.取函数$f(z)=z^{2}\in \mathcal{Z}^{\alpha}$, 显然有$\|f\|_{\mathcal{Z}^{\alpha}}=2$, 则

$\begin{eqnarray*} S_{\varphi}\bigg(\frac{(C_{\phi}^{g}f)''(z)}{C\|f\|_{\mathcal{Z}^{\alpha}}}\bigg) &=&\sup\limits_{z\in \mathbb{D}}(1-|z|^{2})\varphi\bigg(\frac{|\phi(z)g'(z)+\phi'(z)g(z)|}{C}\bigg)\leq1. \end{eqnarray*}$

由此可得

$\begin{eqnarray*} \frac{|\phi'(z)||g(z)|-|\phi(z)||g'(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}<\infty, \end{eqnarray*}$

利用$\phi(z)$的有界性和$k_{3}<\infty$, 即可得到

$\begin{eqnarray} k_{5}=\frac{|\phi'(z)||g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}<\infty. \end{eqnarray}$

(4.3)

对于任何$z\in\mathbb{D}$及非零的$b\in\mathbb{D}$, 令

$\begin{eqnarray*} p_{b}(z)=\frac{(1-|b|^{2})^{2}}{\overline{b}(1-\overline{b}z)^{\alpha}}-\alpha\int_{0}^{z}\frac{1-|b|^{2}}{(1-\overline{b}\lambda)^{\alpha}}d\lambda, \end{eqnarray*}$

则

$\begin{eqnarray*} \sup\limits_{z\in\mathbb{D}}(1-|z|^{2})^{\alpha}|p''_{b}(z)| &\leq&\sup\limits_{z\in\mathbb{D}}(1+|z|)^{\alpha}(1-|z|)^{\alpha}\frac{\alpha(\alpha+1)(1+|b|)^{2}(1-|b|)^{2}|b|}{(1-|z|)^{\alpha}(1-|b|)^{2}}\\ &&+\sup\limits_{z\in\mathbb{D}}(1+|z|)^{\alpha}(1-|z|)^{\alpha}\frac{\alpha^{2}(1+|b|)(1-|b|)|b|}{(1-|z|)^{\alpha}(1-|b|)}\\ &\leq&4\cdot2^{\alpha}\cdot\alpha(\alpha+1)+2\cdot2^{\alpha}\alpha^{2}<\infty, \end{eqnarray*}$

从而$p_{b}\in\mathcal{Z}^{\alpha}$.令$b=\phi(\omega)$, $\omega\in\mathbb{D}$且$\frac{1}{2}<|\phi(\omega)|<1$, 则

$\begin{eqnarray} p'_{\phi(\omega)}(\phi(\omega))=0, p''_{\phi(\omega)}(\phi(\omega))=\frac{\alpha\overline{\phi(\omega)}}{(1-|\phi(\omega)|^{2})^{\alpha}}. \end{eqnarray}$

(4.4)

由$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$的有界性知, 存在一个常数$C$, 使得$\|C_{\phi}^{g}p_{\phi(\omega)}\|_{\mathcal{Z}^{\varphi}}\leq C$, 则由(4.4) 式,

$\begin{eqnarray*} 1&\geq& S_{\varphi}\bigg(\frac{(C_{\phi}^{g}p_{\phi(\omega)})''(z)}{C}\bigg)\geq \sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}(1-|\omega|^{2})\varphi\bigg(\frac{|(C_{\phi}^{g}p_{\phi(\omega)})''(\omega)|}{C}\bigg)\\ &=&\sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}(1-|\omega|^{2})\varphi \bigg(\frac{\alpha|\phi(\omega)||\phi'(\omega)||g(\omega)|}{C(1-|\phi(\omega)|^{2})^{\alpha}}\bigg). \end{eqnarray*}$

由此可得

$\begin{eqnarray} \sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}\frac{|\phi'(\omega)||g(\omega)|}{\varphi^{-1}(\frac{1}{1-|\omega|^{2}})(1-|\phi(\omega)|^{2})^{\alpha}} \leq\sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}\frac{2|\phi(\omega)||\phi'(\omega)||g(\omega)|}{\varphi^{-1}(\frac{1}{1-|\omega|^{2}})(1-|\phi(\omega)|^{2})^{\alpha}}<\infty. \end{eqnarray}$

(4.5)

利用$k_{5}<\infty$, 有

$\begin{eqnarray} \sup\limits_{|\phi(\omega)|\leq\frac{1}{2}}\frac{|\phi'(\omega)||g(\omega)|}{\varphi^{-1}(\frac{1}{1-|\omega|^{2}})(1-|\phi(\omega)|^{2})^{\alpha}} \leq k_{5}\bigg(\frac{4}{3}\bigg)^{\alpha}<\infty. \end{eqnarray}$

(4.6)

则由(4.5) 和(4.6), (4.2) 式成立.

反之, 假设$k_{3}, k_{4}<\infty$, 对于任何$f\in\mathcal{Z}^{\alpha}\backslash\{0\}$, 由引理2.1(ⅰ), 有

$\begin{eqnarray*} &&S_{\varphi}\bigg(\frac{(C_{\phi}^{g}f)''(z)}{C\|f\|_{\mathcal{Z}^{\alpha}}}\bigg)\\ &\leq&\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})\varphi\bigg(\frac{k_{3}\varphi^{-1}(\frac{1}{1-|z|^{2}})|f'(\phi(z))| +k_{4}\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha}|f''(\phi(z))|}{C\|f\|_{\mathcal{Z}^{\alpha}}}\bigg)\\ &\leq&\sup\limits_{z\in\mathbb{D}}(1-|z|^{2})\varphi\bigg(\frac{\frac{2}{1-\alpha}k_{3}+k_{4}}{C}\varphi^{-1}\bigg(\frac{1}{1-|z|^{2}}\bigg)\bigg)\leq1, \\ \end{eqnarray*}$

其中$C\geq\frac{2}{1-\alpha}k_{3}+k_{4}$, 并且利用了事实

$\begin{eqnarray*} \sup\limits_{z\in\mathbb{D}}(1-|\phi(z)|^{2})^{\alpha}|f''(\phi(z))|\leq\|f\|_{\mathcal{Z}^{\alpha}}. \end{eqnarray*}$

故存在一个常数$C$, 使得$\|C_{\phi}^{g}f\|_{\mathcal{Z}^{\varphi}}\leq C\|f\|_{\mathcal{Z}^{\alpha}}$, 即$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界的.

定理4.2  设$g\in H(\mathbb{D})$, $\phi$为$\mathbb{D}$的解析自映射, $0<\alpha<1$.则$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是紧算子当且仅当$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界算子, 且

$\begin{eqnarray} \lim\limits_{|\phi(z)|\rightarrow1}\frac{|\phi'(z)||g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha}}<\infty. \end{eqnarray}$

(4.7)

证 假设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是紧算子, 则显然$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界的.令$\{z_{n}\}_{n\in\mathbb{N}}$是$\mathbb{D}$中的序列使$|\phi(z_{n})|\rightarrow1, $ $n\rightarrow\infty$.令

$\begin{eqnarray*} p_{n}(z)=\frac{(1-|\phi(z_{n})|^{2})^{2}}{\overline{\phi(z_{n})}(1-\overline{\phi(z_{n})}z)^{\alpha}}- \alpha\int_{0}^{z}\frac{1-|\phi(z_{n})|^{2}}{(1-\overline{\phi(z_{n})}\lambda)^{\alpha}}d\lambda, \end{eqnarray*}$

由定理4.1的证明知$\sup\limits_{n\in\mathbb{N}}\|p_{n}\|_{\mathcal{Z}^{\alpha}}<\infty$.同时易见$\{p_{n}\}$在$\mathbb{D}$的紧子集上一致收敛于零, 由引理2.2, $\|C_{\phi}^{g}p_{n}\|_{\mathcal{Z}^{\varphi}}\rightarrow0, n\rightarrow\infty$.则

$\begin{eqnarray*} 1\geq S_{\varphi}\bigg(\frac{(C_{\phi}^{g}p_{n})''(z_{n})}{\|C_{\phi}^{g}p_{n}\|_{\mathcal{Z}^{\varphi}}}\bigg) \geq(1-|z_{n}|^{2})\varphi \bigg(\frac{\alpha|\phi(z_{n})||\phi'(z_{n})||g(z_{n})|}{(1-|\phi(z_{n})|^{2})^{\alpha}\|C_{\phi}^{g}p_{n}\|_{\mathcal{Z}^{\varphi}}}\bigg), \end{eqnarray*}$

由此可得

$\begin{eqnarray*} \frac{|\phi(z_{n})||\phi'(z_{n})||g(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha}}\leq\frac{1}{\alpha}\|C_{\phi}^{g}p_{n}\|_{\mathcal{Z}^{\varphi}}. \end{eqnarray*}$

故

$\begin{eqnarray*} \lim\limits_{|\phi(z_{n})|\rightarrow1}\frac{|\phi'(z_{n})||g(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha}}= \lim\limits_{n\rightarrow\infty}\frac{|\phi(z_{n})||\phi'(z_{n})||g(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha}}=0. \end{eqnarray*}$

反之, 假设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$有界且(4.7) 式成立, 则由定理4.1, 可得$k_{3}, k_{5}<\infty$, 且对于任意的$\epsilon>0$, 存在$\delta\in(0, 1)$, 使得只要$\delta<|\phi(z)|<1$, 就有

$\begin{eqnarray} \frac{|\phi'(z)||g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha}}<\epsilon. \end{eqnarray}$

(4.8)

令$\{f_{n}\}_{n\in\mathbb{N}}$为$\mathcal{Z}^{\alpha}$中的任意有界序列, $\sup\limits_{n\in\mathbb{N}}\|f_{n}\|_{\mathcal{Z}^{\alpha}}\leq L$, 且在$\mathbb{D}$的紧子集上一致收敛于零, 令$K=\{z\in\mathbb{D}:|\phi(z)|\leq\delta\}$, 又$(C_{\phi}^{g}f_{n})(0)=0$, 则由$k_{3}, k_{5}<\infty$及(4.8) 式可得

$\begin{eqnarray*} \|C_{\phi}^{g}f_{n}\|_{\mathcal{Z}^{\varphi}} &\leq&|f'_{n}(\phi(0))||g(0)| +\sup\limits_{z\in\mathbb{D}}\frac{|g'(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}|f'_{n}(\phi(z))| +\sup\limits_{z\in K}\frac{|\phi'(z)||g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}|f''_{n}(\phi(z))|\\ &&+\sup\limits_{z\in \mathbb{D}\backslash K}\frac{|\phi'(z)||g(z)|(1-|\phi(z)|^{2})^{\alpha}|f''_{n}(\phi(z))|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha}}\\ &\leq&|f'_{n}(\phi(0))||g(0)| +k_{3}\sup\limits_{z\in\mathbb{D}}|f'_{n}(\phi(z))|+k_{5}\sup\limits_{|\omega|\leq\delta}|f''_{n}(\omega)|+L\epsilon. \end{eqnarray*}$

由于$\{f_{n}\}$在$\mathbb{D}$的紧子集上一致收敛于零, 则由引理2.3知$ \lim\limits_{n\rightarrow\infty}\sup\limits_{z\in\mathbb{D}}|f'_{n}(\phi(z))|=0. $再由Cauchy估计, $\{f'_{n}\}$, $\{f''_{n}\}$也在$\mathbb{D}$的紧子集上一致收敛于零.特别地, $\{\phi(0)\}$, $\{\omega:|\omega|\leq\delta\}$均为$\mathbb{D}$的紧子集, 则有$ \lim\limits_{n\rightarrow\infty}|f'_{n}(\phi(0))|=0, \lim\limits_{n\rightarrow\infty}\sup\limits_{|\omega|\leq\delta}|f''_{n}(\omega)|=0.$从而由引理2.2可得$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是紧算子.

定理4.3  设$g\in H(\mathbb{D})$, $\phi$为$\mathbb{D}$的解析自映射, $\alpha>1$.则$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界算子当且仅当

$k_{6}=\sup\limits_{z\in\mathbb{D}}\frac{|g'(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha-1}}<\infty,$

(4.9)

$k_{7}=\sup\limits_{z\in\mathbb{D}}\frac{|\phi'(z)||g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha}}<\infty.$

(4.10)

证 假设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界算子, 利用定理4.1, 可证得$k_{3}<\infty$, 通过取函数$p_{b}(z)$, 进行相似的讨论, 即可证得(4.10) 式.下证(4.9) 式, 对于任何$z\in\mathbb{D}$, 及非零的$b\in\mathbb{D}$, 令

$\begin{eqnarray*} q_{b}(z)=\frac{(1-|b|^{2})^{2}}{\overline{b}(1-\overline{b}z)^{\alpha}}, \end{eqnarray*}$

则易证$q_{b}\in\mathcal{Z}^{\alpha}$.令$b=\phi(\omega), \omega\in\mathbb{D}$且$\frac{1}{2}<|\phi(\omega)|<1$, 直接计算可得

$\begin{eqnarray} q'_{\phi(\omega)}(\phi(\omega))=\frac{\alpha}{(1-|\phi(\omega)|^{2})^{\alpha-1}}, q''_{\phi(\omega)}(\phi(\omega))=\frac{\alpha(\alpha+1)\overline{\phi(\omega)}}{(1-|\phi(\omega)|^{2})^{\alpha}}. \end{eqnarray}$

(4.11)

另外由$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$的有界性知, 存在一个常数$C$, 使得$\|C_{\phi}^{g}q_{\phi(\omega)}\|_{\mathcal{Z}^{\varphi}}\leq C$, 则由(4.11) 式,

$\begin{eqnarray*} 1&\geq& S_{\varphi}\bigg(\frac{(C_{\phi}^{g}q_{\phi(\omega)})''(z)}{C}\bigg)\geq \sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}(1-|\omega|^{2})\varphi\bigg(\frac{|(C_{\phi}^{g}q_{\phi(\omega)})''(\omega)|}{C}\bigg)\\ &=&\sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}(1-|\omega|^{2})\varphi \bigg(\frac{\big|\frac{\alpha}{(1-|\phi(\omega)|^{2})^{\alpha-1}}g'(\omega)+ \frac{\alpha(\alpha+1)\overline{\phi(\omega)}}{(1-|\phi(\omega)|^{2})^{\alpha}}\phi'(\omega)g(\omega)\big|}{C}\bigg)\\ &\geq&\sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}(1-|\omega|^{2})\varphi \bigg(\frac{\alpha|g'(\omega)|}{C(1-|\phi(\omega)|^{2})^{\alpha-1}}- \frac{\alpha(\alpha+1)|\phi(\omega)||\phi'(\omega)||g(\omega)|}{C(1-|\phi(\omega)|^{2})^{\alpha}}\bigg). \end{eqnarray*}$

利用$k_{7}<\infty$可得

$\begin{eqnarray} \sup\limits_{\frac{1}{2}<|\phi(\omega)|<1}\frac{|g'(\omega)|}{\varphi^{-1}(\frac{1}{1-|\omega|^{2}})(1-|\phi(\omega)|^{2})^{\alpha-1}}\leq C+(\alpha+1)k_{7}<\infty. \end{eqnarray}$

(4.12)

利用$k_{3}<\infty$, 我们有

$\begin{eqnarray} \sup\limits_{|\phi(\omega)|\leq\frac{1}{2}}\frac{|g'(\omega)|}{\varphi^{-1}(\frac{1}{1-|\omega|^{2}})(1-|\phi(\omega)|^{2})^{\alpha-1}}\leq k_{3}\bigg(\frac{4}{3}\bigg)^{\alpha-1}. \end{eqnarray}$

(4.13)

由(4.12) 和(4.13) 式即可得到(4.9) 式.

反之, 假设$k_{6}, k_{7}<\infty$, 根据引理2.1(iii), 仿照定理4.1的方法, 不难证得$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界算子.

定理4.4 设$g\in H(\mathbb{D})$, $\phi$为$\mathbb{D}$的解析自映射, $\alpha>1$.则$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是紧算子当且仅当$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界算子, 且

$\lim\limits_{|\phi(z)|\rightarrow1}\frac{|g'(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha-1}}<\infty,$

(4.14)

$\lim\limits_{|\phi(z)|\rightarrow1}\frac{|\phi'(z)||g(z)|}{\varphi^{-1}(\frac{1}{1-|z|^{2}})(1-|\phi(z)|^{2})^{\alpha}}<\infty.$

(4.15)

证 假设$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是紧算子, 则显然$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是有界的.令$\{z_{n}\}_{n\in\mathbb{N}}$是$\mathbb{D}$中的序列使$|\phi(z_{n})|\rightarrow1$ ($n\rightarrow\infty$).则利用定理4.2, 通过取函数$p_{n}(z)$, 可证得(4.15) 式成立.下证(4.14) 式, 令

$\begin{eqnarray*} q_{n}(z)=\frac{(1-|\phi(z_{n})|^{2})^{2}}{\overline{\phi(z_{n})}(1-\overline{\phi(z_{n})}z)^{\alpha}}, \end{eqnarray*}$

则$\sup\limits_{n\in\mathbb{N}}\|q_{n}\|_{\mathcal{Z}^{\alpha}}<\infty$且$\{q_{n}\}$在$\mathbb{D}$的紧子集上一致收敛于零, 由引理2.2, $\|C_{\phi}^{g}q_{n}\|_{\mathcal{Z}^{\varphi}}\rightarrow0, n\rightarrow\infty$.则有

$\begin{eqnarray*} 1&\geq& S_{\varphi}\bigg(\frac{(C_{\phi}^{g}q_{n})''(z_{n})}{\|C_{\phi}^{g}q_{n}\|_{\mathcal{Z}^{\varphi}}}\bigg)\\ &\geq&(1-|z_{n}|^{2})\varphi \bigg(\frac{\big|\frac{\alpha g'(z_{n})}{(1-|\phi(z_{n})|^{2})^{\alpha-1}}+ \frac{\alpha(\alpha+1)\overline{\phi(z_{n})}\phi'(z_{n})g(z_{n})}{(1-|\phi(z_{n})|^{2})^{\alpha}}\big|}{\|C_{\phi}^{g}q_{n}\|_{\mathcal{Z}^{\varphi}}}\bigg)\\ &\geq&(1-|z_{n}|^{2})\varphi \bigg(\frac{\alpha|g'(z_{n})|}{\|C_{\phi}^{g}q_{n}\|_{\mathcal{Z}^{\varphi}}(1-|\phi(z_{n})|^{2})^{\alpha-1}}- \frac{\alpha(\alpha+1)|\phi(z_{n})||\phi'(z_{n})||g(z_{n})|}{\|C_{\phi}^{g}q_{n}\|_{\mathcal{Z}^{\varphi}}(1-|\phi(z_{n})|^{2})^{\alpha}}\bigg). \end{eqnarray*}$

由此可得

$\begin{eqnarray} \frac{|g'(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha-1}}\leq \frac{1}{\alpha}\|C_{\phi}^{g}q_{n}\|_{\mathcal{Z}^{\varphi}}+ \frac{(\alpha+1)|\phi(z_{n})||\phi'(z_{n})||g(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha}}, \end{eqnarray}$

(4.16)

利用(4.15), (4.16) 式即可得

$\begin{eqnarray*} \lim\limits_{|\phi(z_{n})|\rightarrow1}\frac{|g'(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha-1}}= \lim\limits_{n\rightarrow\infty}\frac{|g'(z_{n})|}{\varphi^{-1}(\frac{1}{1-|z_{n}|^{2}})(1-|\phi(z_{n})|^{2})^{\alpha-1}}=0. \end{eqnarray*}$

反之, 假设(4.14), (4.15) 式成立, 则利用引理2.1(iii), 仿照定理4.2和定理3.3的过程, 易证$C_{\phi}^{g}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\varphi}$是紧算子.